10th Polynomials-Project Works

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Polynomial Projects - Student Submission

Polynomial Projects

Mathematics Project Work - Class X

Name	Rahul Sharma
Class	X - A
Roll Number	25
Lesson	Polynomials
Academic Year	2023-2024

Project 1: Exploring the Parabola - A Graphical Journey into Quadratic Polynomials

Aim

To understand the relationship between the coefficients of a quadratic polynomial (ax² + bx + c) and the shape/position of its graph (a parabola), and to visually interpret its zeroes.

Objectives

To plot graphs of various quadratic polynomials
To analyze the effect of coefficients on the parabola's shape and position
To identify zeroes of polynomials from their graphs
To verify the relationship between zeroes and coefficients

Materials

Graph paper
Pencil and eraser
Scale/Ruler
Different colored pens

Tools

Mathematical compass (optional)
Calculator

Procedure

Selected various quadratic polynomials with different coefficient values
Created value tables for each polynomial by substituting x values
Plotted points on graph paper for each polynomial using appropriate scale
Joined the points with smooth curves to form parabolas
Analyzed the direction, width, and position of each parabola
Identified x-intercepts (zeroes) from the graphs
Verified the zeroes using algebraic methods
Compared the sum and product of zeroes with coefficient relationships

Sample Graph: y = x² + 2x - 3

[Graph showing parabola with vertex at (-1,-4) and zeroes at x=-3 and x=1]

Observations

When a > 0, the parabola opens upwards; when a < 0, it opens downwards
Larger |a| values produce narrower parabolas
The constant term c determines the y-intercept of the graph
The coefficient b affects the position of the vertex along the x-axis
The number of x-intercepts (zeroes) can be 0, 1, or 2
For y = x² + 2x - 3, the zeroes are x = -3 and x = 1
Sum of zeroes = -2 = -b/a, Product of zeroes = -3 = c/a

Polynomial	Zeroes	Sum of Zeroes	Product of Zeroes
x² + 2x - 3	-3, 1	-2	-3
x² - 5x + 6	2, 3	5	6
2x² - 8x + 6	1, 3	4	3

Conclusion

The graph of a quadratic polynomial is always a parabola. The coefficient a determines the direction and width of the parabola, b affects the position of the vertex, and c gives the y-intercept. The zeroes of the polynomial are the x-coordinates where the graph intersects the x-axis. The relationships α+β = -b/a and αβ = c/a were verified through this project.

Experience

This project helped me visualize the abstract concept of polynomials. Drawing the graphs made it easier to understand how changing coefficients affects the shape and position of the parabola. I learned to connect algebraic expressions with their geometric representations. The hands-on approach made the mathematical relationships more memorable and intuitive.

Quadratic Polynomial Calculator

Coefficient a:

Coefficient b:

Coefficient c:

Project 2: The Algebra-Geometry Bridge: Zeroes and Coefficients of Cubic Polynomials

Aim

To verify the relationship between the zeroes and coefficients of a cubic polynomial and to use the Division Algorithm to find all zeroes when one is known.

Objectives

To verify relationships between zeroes and coefficients of cubic polynomials
To apply the Division Algorithm to factorize polynomials
To find all zeroes of a cubic polynomial when one zero is known
To connect algebraic and geometric representations of polynomials

Materials

Pen and paper
Calculator
Chart paper for presentation

Tools

Mathematical tables (if needed)

Procedure

Selected cubic polynomials that can be factorized easily
Found zeroes of each polynomial by factorization method
Recorded the zeroes and coefficients for each polynomial
Verified the relationships:
- α + β + γ = -b/a
- αβ + βγ + γα = c/a
- αβγ = -d/a
Selected a cubic polynomial with one known zero
Applied the Division Algorithm to divide the polynomial by (x - known zero)
Factorized the quotient to find the remaining zeroes
Verified all zeroes by substituting in the original polynomial

Cubic Polynomial Graph: y = x³ - 4x

[Graph showing cubic curve with zeroes at x=-2, x=0, and x=2]

Observations

Cubic polynomials can have 1 or 3 real zeroes
The relationships between zeroes and coefficients hold for all cubic polynomials
The Division Algorithm is an efficient method to find all zeroes when one is known
The graph of a cubic polynomial can intersect the x-axis at 1 or 3 points
For x³ - 4x, the zeroes are -2, 0, and 2
Sum of zeroes = 0 = -b/a, Sum of products = -4 = c/a, Product = 0 = -d/a

Polynomial	Zeroes (α, β, γ)	α+β+γ	αβ+βγ+γα	αβγ
x³ - 4x	-2, 0, 2	0	-4	0
x³ - 6x² + 11x - 6	1, 2, 3	6	11	6
2x³ - 5x² - 14x + 8	4, -2, ½	2.5	-7	-4

Conclusion

Cubic polynomials follow specific relationships between their zeroes and coefficients. The Division Algorithm provides a systematic method to find all zeroes when one zero is known. These relationships and methods connect the algebraic and geometric aspects of polynomials, enhancing our understanding of their behavior and properties.

Experience

Working with cubic polynomials helped me understand higher-degree algebraic expressions. The Division Algorithm was particularly interesting as it provided a structured approach to polynomial factorization. Seeing the consistent relationships between zeroes and coefficients across different examples reinforced my understanding of polynomial theory. This project strengthened my algebraic manipulation skills and deepened my appreciation for the interconnectedness of mathematical concepts.

Cubic Polynomial Calculator

Coefficient a:

Coefficient b:

Coefficient c:

Coefficient d:

Acknowledgement

I would like to express my sincere gratitude to my mathematics teacher, Mrs. Priya Mehta, for her guidance and support throughout this project. Her explanations and insights helped me understand the concepts clearly. I also thank my school for providing the necessary resources and my parents for their encouragement. Finally, I acknowledge the textbook authors whose work formed the foundation of this project.

References

Mathematics Textbook for Class X - NCERT
Comprehensive Mathematics - R.D. Sharma
Mathematics Class X - R.S. Aggarwal
Khan Academy - Online resources on polynomials
Byju's Learning App - Interactive polynomial lessons

10th Sets-Project Works

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Sets Projects - Mathsstudent.com

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Venn Diagrams and Set Operations-Project-1

Name: K. Syam

Class: X

Roll Number: 10

Lesson: Sets

Subject: Mathematics

Aim

To understand and visualize set operations using Venn diagrams and to solve practical problems using set theory concepts.

Objectives

To understand the concept of sets, subsets, and universal sets
To learn and apply set operations: union, intersection, and difference
To represent sets and set operations using Venn diagrams
To solve real-world problems using set theory

Materials & Tools

Materials Used: Chart paper, colored pens, ruler, compass, scissors

Tools: Mathematical calculation, Venn diagram representation

Procedure

Collected data from classmates about their preferences for different subjects
Defined universal set as all students in the class
Created sets for students liking Mathematics, Science, and English
Drew Venn diagrams to represent the relationships between these sets
Calculated union, intersection, and difference of the sets
Analyzed the results to find patterns and relationships

Observations & Data Analysis

Survey Data Collected:

Total students in class: 40

Students who like Mathematics: 25
Students who like Science: 22
Students who like English: 18
Students who like both Mathematics and Science: 12
Students who like both Mathematics and English: 8
Students who like both Science and English: 6
Students who like all three subjects: 4

Set Operations:

Let M = Set of students who like Mathematics

Let S = Set of students who like Science

Let E = Set of students who like English

Set Operation	Calculation	Result
M ∪ S (Union)	n(M) + n(S) - n(M∩S) = 25 + 22 - 12	35
M ∪ E (Union)	n(M) + n(E) - n(M∩E) = 25 + 18 - 8	35
S ∪ E (Union)	n(S) + n(E) - n(S∩E) = 22 + 18 - 6	34
M ∩ S (Intersection)	Given in survey	12
M ∩ E (Intersection)	Given in survey	8
S ∩ E (Intersection)	Given in survey	6
M ∩ S ∩ E (Intersection)	Given in survey	4

Conclusion

Through this project, I learned that Venn diagrams are powerful visual tools for understanding set relationships. The survey data analysis showed that many students have overlapping interests in different subjects. Set operations like union, intersection, and difference helped quantify these relationships. The principle of inclusion-exclusion was particularly useful in calculating union of sets without double-counting common elements.

Experience of the Student

This project helped me understand how abstract mathematical concepts like sets have practical applications in organizing and analyzing real-world data. Creating Venn diagrams made the relationships between sets much clearer than just working with formulas. I also learned how to conduct a survey and present the results mathematically.

Acknowledgement

1. I thank our Mathematics teacher for guiding us through the concepts of set theory

2. I appreciate my classmates for actively participating in the survey

Reference Books / Resources

X Class State Board Mathematics Textbook - Chapter 2: Sets
IX Class Mathematics Textbook
Online resources about Venn diagrams and set operations

Mathsstudent.com

Finite and Infinite Sets in Daily Life- Project -2

Name: K. Smalhor2

Class: X

Roll Number: 10

Lesson: Sets

Subject: Mathematics

Aim

To identify and classify various collections as finite or infinite sets and to understand the concept of cardinality in finite sets.

Objectives

To differentiate between finite and infinite sets
To identify empty sets and understand their properties
To calculate cardinality of finite sets
To find subsets of given finite sets
To apply set concepts to real-life collections

Materials & Tools

Materials Used: Notebook, pen, ruler, collection of various objects

Tools: Observation, classification, mathematical calculation

Procedure

Collected various groups of objects from daily life
Classified each collection as a set
Determined whether each set is finite or infinite
For finite sets, calculated their cardinality
Found all possible subsets of smaller finite sets
Identified empty sets in practical situations

Observations & Analysis

Classification of Sets:

Set Description	Roster Form	Type	Cardinality
Days of the week	{Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}	Finite	7
Prime numbers less than 20	{2, 3, 5, 7, 11, 13, 17, 19}	Finite	8
Vowels in English alphabet	{a, e, i, o, u}	Finite	5
Natural numbers	{1, 2, 3, 4, ...}	Infinite	Not defined
Points on a line segment	Cannot be listed	Infinite	Not defined
Even prime numbers greater than 2	{ }	Empty	0

Subsets of a Finite Set:

Let A = {1, 2, 3}

All subsets of A:

Empty set: { } or ∅
Single element subsets: {1}, {2}, {3}
Two element subsets: {1, 2}, {1, 3}, {2, 3}
The set itself: {1, 2, 3}

Total number of subsets = 2³ = 8

Cardinality of Union and Intersection:

Let B = {2, 4, 6, 8, 10} and C = {1, 2, 3, 4, 5}

n(B) = 5, n(C) = 5
B ∩ C = {2, 4}, so n(B ∩ C) = 2
B ∪ C = {1, 2, 3, 4, 5, 6, 8, 10}, so n(B ∪ C) = 8
Verification: n(B ∪ C) = n(B) + n(C) - n(B ∩ C) = 5 + 5 - 2 = 8

Conclusion

This project helped me understand the classification of sets as finite, infinite, or empty. Finite sets have a definite number of elements that can be counted, while infinite sets continue indefinitely. The empty set is a special case with no elements. The concept of cardinality is applicable only to finite sets, and the formula for the number of subsets (2ⁿ where n is the cardinality) is a powerful tool in set theory. These concepts have practical applications in data organization, database management, and problem-solving.

Experience of the Student

Working on this project made me more observant about collections in daily life. I started seeing sets everywhere - from utensils in the kitchen to books on my shelf. Calculating all subsets of a set was challenging but interesting, and I discovered the pattern that the number of subsets doubles with each additional element. The concept of infinite sets was initially difficult to grasp, but thinking about examples like the set of all points on a line helped me understand it better.

Acknowledgement

1. I thank our Mathematics teacher for explaining the concepts of finite and infinite sets clearly

2. I appreciate my family members for helping me collect various objects for classification

Reference Books / Resources

X Class State Board Mathematics Textbook - Chapter 2: Sets
IX Class Mathematics Textbook
Online resources about finite and infinite sets

10th Real Numbers- Project Works

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Real Numbers Projects - Mathsstudent.com

Mathsstudent.com

Euclid's Algorithm in Action!

Name: K. Arun

Class: X

Roll Number: 10

Lesson: Real Numbers

Subject: Mathematics

Aim

To find the Highest Common Factor (HCF) of given pairs of numbers using Euclid's Division Algorithm and to verify it through a practical activity.

Objectives

To understand and apply Euclid's Division Lemma
To find HCF using Euclid's Algorithm
To verify the algorithm with a practical activity using paper strips

Materials & Tools

Materials Used: Paper strips (two different colours), scale, pen, pencil, sketch pens, scissors

Tools: Mathematical calculation, Practical experiment

Procedure

Two pairs of numbers were selected: (50, 75) and (32, 96)
For each pair, Euclid's Division Algorithm was applied step-by-step until the remainder became zero
To verify the result for (50, 75), two paper strips of lengths 50 cm and 75 cm were cut
The longer strip (75 cm) was measured using the shorter strip (50 cm)
The leftover part (25 cm) was used to measure the 50 cm strip
The process continued until no remainder was left

Observations & Calculations

A) For numbers 50 and 70:

70 = 50 × 1 + 20
50 = 20 × 2 + 10
20 = 10 × 2 + 0
HCF (50, 70) = 10

B) For numbers 32 and 96:

96 = 32 × 3 + 0
HCF (32, 96) = 32

Application of Euclid's Algorithm

Numbers (a, b)	Division Steps (a = bq + r)	HCF
50, 70	70 = 50 × 1 + 20 50 = 20 × 2 + 10 20 = 10 × 2 + 0	10
32, 96	96 = 32 × 3 + 0	32
1860, 2015	2015 = 1860 × 1 + 155 1860 = 155 × 12 + 0	155

Conclusion

Euclid's Division Algorithm provides a systematic and efficient method for finding the HCF of two positive integers. The practical activity with paper strips visually confirms that the HCF is the largest common length that can measure both given lengths without any remainder.

Experience of the Student

This project helped me understand that mathematics is not just about calculations but also about visual and practical understanding. Cutting the strips and physically measuring them made the concept of HCF very clear and interesting.

Acknowledgement

1. My sincere thanks to our Mathematics teacher for guiding us through this project

2. I would also like to thank my classmates for their cooperation and support

Reference Books / Resources

X Class State Board Mathematics Textbook
IX Class Mathematics Textbook

Mathsstudent.com

Project Work-2 Decoding Decimals: Terminating or Non-Terminating?

Name: R.Reema

Class: X

Roll Number: 10

Lesson: Real Numbers

Subject: Mathematics

Aim

To investigate the relationship between the prime factors of the denominator of a rational number and the nature of its decimal expansion (terminating or non-terminating repeating).

Objectives

Expressing rational numbers in their decimal form
Expressing the denominator of the rational number in its prime factorized form
Differentiating between terminating and non-terminating recurring decimals

Materials & Tools

Materials Used: Pen, Pencil, Scale, Chart Paper, Coloured Pens

Tools: Collection of rational numbers, Mathematical calculation, Prime factorization

Procedure

Collected various rational numbers
Each rational number was converted into its decimal form by performing division
Noted whether the decimal was Terminating (T) or Non-Terminating Repeating (NR)
Each rational number was written in its simplest form p/q
The denominator (q) was factorized into its prime factors
The observations were recorded in a table for analysis

Observations

Relation between Denominator and Decimal Expansion

S.No.	Rational Number	Decimal Form	Type	Simplified Form	Prime Factors of q	Observation
1	3/8	0.375	T	3/8	2³	Only prime factor 2
2	7/25	0.28	T	7/25	5²	Only prime factor 5
3	13/125	0.104	T	13/125	5³	Only prime factor 5
4	9/20	0.45	T	9/20	2² × 5¹	Prime factors 2 and 5
5	1/3	0.333...	NR	1/3	3¹	Prime factor 3
6	5/12	0.41666...	NR	5/12	2² × 3¹	Has prime factor 3
7	7/13	0.538461...	NR	7/13	13¹	Prime factor 13
8	29/343	0.08454...	NR	29/343	7³	Prime factor 7

Conclusion

From the observations, it is concluded that the decimal expansion of a rational number p/q (in simplest form) terminates if and only if the prime factorization of the denominator q is of the form 2ⁿ × 5ᵐ, where n and m are non-negative integers. If the denominator has any prime factor other than 2 or 5, the decimal expansion is non-terminating and repeating.

Result

A rational number p/q, where p and q are co-prime, has a terminating decimal expansion if the prime factors of q are only 2 and/or 5.

Experience of the Student

While working on this project, I learned to predict the nature of a decimal just by looking at the denominator of a fraction. Initially, I thought all decimals either stop or show a simple pattern, but I discovered that some repeats after a long sequence, which was fascinating.

Acknowledgement

1. I thank our Mathematics teacher for her invaluable guidance

2. I extend my gratitude to my parents for their support

Reference Books / Resources

X Class State Board Mathematics Textbook (Chapter 1: Real Numbers)
IX Class Mathematics Textbook

Maths Project Works

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Maths Project works for all classes

8th Class Maths Project Works

Rational Numbers TM	Click Here to Download
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Square Roots and Cube Roots TM	Click Here to Download
Frequency Distribution Tables and Graphs TM	Click Here to Download
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Area of Plane Figures TM	Click Here to Download
Direct and Inverse Propartions TM	Click Here to Download
Algebraic Expressions TM	Click Here to Download
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